Large-Scale and Parallel Matrix Computations in Linear Control

这个跨学科项目将致力于研究线性控制系统设计和分析中出现的几个基本线性代数问题的大规模解决和并行计算的有效数值方法。将特别关注计算上可行的算法的开发和实现：大型二阶模型的反馈稳定；计算大型稀疏矩阵的频率响应和h∞范数；以及重要矩阵方程的并行和高性能计算，如Lyapunov， Sylvester和Riccati以及特征值分配问题。此外，本文还考虑了数值分析的一些方面，如问题的调节和算法的稳定性。将强调在矩阵计算中使用最先进的技术，如GMRES和QMR算法，揭示秩的QR分解，条件估计方案，如SPICE， ICE， ACE等，以及相关软件包（如LAPACK）。本项目将开发的算法及其在矢量超级计算机和大规模并行架构上的实现，将允许快速、准确和可靠地解决实际情况中出现的大规模控制问题，如大空间结构的设计、电力系统的控制，以及其他无法在合理时间内处理的问题。如果有的话。因此，拟议的研究将解决最近小组报告“控制理论的未来方向：数学视角”中概述的控制理论中大规模并行计算领域研究的一些关键需求。这个项目很及时，对应用数学家、控制理论家和工程师都有好处。

This interdisciplinary project will be devoted to the study of effective numerical methods for large-scale solutions and parallel computations of several fundamental linear algebra problems arising in the design and analysis of linear control systems. Specific attention will be paid to the development and implementation of computationally viable algorithms for: feedback stabilization of the large second-order model; computing the frequency response and the H-infinity norm of a large and sparse matrix; and parallel and high performance computing of important matrix equations such as the Lyapunov, Sylvester, and Riccati and the eigenvalue assignment problems. Some numerical analysis aspects such as conditioning of the problems and stability of the algorithms will be considered, as well. The use of state-of-the-art techniques in matrix computations such as the GMRES and QMR algorithms, Rank Revealing QR factorizations, condition estimations schemes such as SPICE, ICE, ACE, etc., and the associated software packages (e.g. LAPACK) will be emphasized. The algorithms to be developed in this project and their implementations on vector supercomputer and massive parallel architectures will permit fast, accurate, and reliable solutions of large-scale control problems arising in practical situations such as the design of large space structures, control of power systems, and for other problems that otherwise could not be handled in a reasonable time, if at all. The proposed research will thus address some of the critical needs for research in the area of large-scale and parallel computations in control theory outlined in the recent panel report "Future Directions in Control Theory: A Mathematical Perspective". The project is timely and benefit both applied mathematicians and control theorists and engineers.